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 A276323 a(n) = (binomial(2 * prime(n + 3), prime(n + 3)) * A005259(prime(n + 3) - 1) - 2)/prime(n + 3)^5 for n >= 1. 2
 4382314, 59821998476834, 338197165389273486, 17314015796594772560245514, 145853326344012138627669357202, 12936469013977571458378002436843685186, 15931675838688077485749893663903436780403973163302 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Let p be a prime > 5. Binomial(2 * p, p) * A005259(p - 1) == 2 (mod p^5). So a(n) is an integer. LINKS Seiichi Manyama, Table of n, a(n) for n = 1..88 Julian Rosen, Periods, supercongruences, and their motivic lifts, arXiv:1608.06864 [math.NT], 2016. EXAMPLE a(1) = (binomial(14, 7) * A005259(6) - 2)/7^5 = (3432 * 21460825 - 2)/7^5 = 4382314. MATHEMATICA Table[(Binomial[2 Prime[n + 3], Prime[n + 3]] Sum[(Binomial[#, k] Binomial[# + k, k])^2, {k, 0, #}] &[Prime[n + 3] - 1] - 2)/Prime[n + 3]^5, {n, 7}] (* Michael De Vlieger, Aug 30 2016 *) PROG (Ruby) require 'prime' def C(n, r) r = [r, n - r].min return 1 if r == 0 return n if r == 1 numerator = (n - r + 1..n).to_a denominator = (1..r).to_a (2..r).each{|p| pivot = denominator[p - 1] if pivot > 1 offset = (n - r) % p (p - 1).step(r - 1, p){|k| numerator[k - offset] /= pivot denominator[k] /= pivot } end } result = 1 (0..r - 1).each{|k| result *= numerator[k] if numerator[k] > 1 } return result end def A005259(n) i = 0 a, b = 1, 5 ary = [1] while i < n i += 1 a, b = b, ((((34 * i + 51) * i + 27) * i + 5) * b - i ** 3 * a) / (i + 1) ** 3 ary << a end ary end def A276323(n) p_ary = Prime.take(n + 3)[3..-1] a = A005259(p_ary[-1] - 1) ary = [] p_ary.each{|i| j = C(2 * i, i) * a[i - 1] - 2 break if j % i ** 5 > 0 ary << j / i ** 5 } ary end CROSSREFS Cf. A000984, A005259. Sequence in context: A254259 A254305 A237537 * A288086 A210297 A019288 Adjacent sequences: A276320 A276321 A276322 * A276324 A276325 A276326 KEYWORD nonn AUTHOR Seiichi Manyama, Aug 30 2016 STATUS approved

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Last modified August 13 05:34 EDT 2024. Contains 375113 sequences. (Running on oeis4.)